Signal, detection and estimation using a hybrid quantum circuit

We investigate a hybrid device allowing a photon–phonon coupling of a transmission line radiation (TLR) and a nanoeletromechanical system (NEMS), mediated by a superconducting qubit population imbalance. We demonstrate the derivation of an effective Hamiltonian for the strongly dispersive regime for this system. The qubit works as a quantum switch, allowing a conditioned transfer of excitations between the TLR and NEMS. We show that this regime allows the system to be employed for signal processing and force estimation. Additionally, we explore the ability of the quantum switch to generate non-classical states.


Physical system
The circuit under study consists of a transmission line resonator (TLR) capacitively coupled to a NEMS through a charge qubit, containing a single Cooper pair box (CPB) made of a superconducting island connected to a large reservoir through two Josephson tunnel junctions with Josephson energy and capacitance E J and C J , respectively.The Cooper pair box is capacitively coupled to the TLR and the NEMS with gate capacitances C t and C n , respectively (see Fig. 1).In the Cooper-pairs number basis, the system Hamiltonian can be written as 14,15 where E c = e/2C is the single electron charging energy (e is the electronic charge), C = C J + C t + C n ≈ C J + C t is its total capacitance 2 , n represents the expected Cooper pair number on the island, and n g is the gate charge applied to it.Following the description, for n = 0, 1 , (associated with the accumulation of a single Cooper pair on the island-a current assumption when the superconducting system is at the point of charge degeneracy for states {|0�, |1�} , where any transition to states of higher excitation is suppressed 2 , see Fig. 2), the Hamiltonian (1) is rewritten as where By changing the base of the CPB to and (2) The circuit schematized on a platform contains an electromagnetic shield with two grounded lines, and between them is a quantum hybrid circuit composed of the capacitive coupling between a TLR and a suspended NEMS mediated by a qubit superconducting 2,5 .The TLR also has a channel capacitively coupled to its output that connects a beam splitter (BS) with two linear detectors for IQ mix measurements 16,17 .

Figure 2.
Representation of qubit energy levels as a function of gate charge, n g , can be rewritten as a two- level quantum system with transition energy E J and hybrid coupling energy, E 0 −E 1 2 , (light-matter-mechanical oscillator).where E 1 − E 0 = 4E c 1 − 2n g , σx = g �e| + |e� g and σz = g g − |e��e| .From now on we call it the qubit base (See Fig. 2).Now we proceed to the proper treatment for the interaction between qubit-TLR and qubit-NEMS by extending the quantization to the gate charge n g → n 0 + δ nt + δ nn , where 2,6,18,19 is the the bias gate composed of a DC field V 0 in TLR, the is the induced charge due to the TLR 14 , being V (t) = ω t Lc âe −iω t t + â † e iω t t , the voltage at the center of the TLR of length L, and capacitance density c.Here, ω t and â(â † ) are the frequency and annihilation (creation) operator of the TLR, respectively, and represents the NEMS flexural mode shift, with V n is the potential difference that can be applied to the NEMS, the capacitance formed between the NEMS and the qubit can be expressed and approximated as with ε o is Vacuum permittivity, A is the rara of the lateral longitudinal section of the NEMS, and d is the equilibrium distance between the NEMS and the qubit.Therefore, using only the first-order term, we obtain 20 where x = 2mω n be −iω n t + b † e iω n t representing the quantised bending mode of NEMS (equivalent to a quantum harmonic oscillator), with ω n , m, dC n dx ≈ ε o A d 2 and b( b † ) being the frequency, mass, first order coefficient of the capacitance expansion C n ≡ C n (x) , and the NEMS annihilation (creation) operator, respectively.Now, to better represent Hamiltonian (3), we rewrite with n o = 0.5, dC N dx V N , and ω q = E J .Note that is manipulated by V N , and ω q is manipulated by classical flux applied to the qubit loop 6 .In the interaction picture obtained by the unitary transformation Ût = e −i Ĥo t/ , where Ĥo = ω q σz /2 , corresponding to the qubit free evolution, the Hamiltonian becomes (with σ+ = |e� g and σ− = g �e|) The overall dynamics could be analyzed from this Hamiltonian, however, we can keep only the more relevant terms in the regime in which ω t + ω q ≫ |ω t − ω q | , ω n + ω q ≫ |ω n − ω q | , the rotating wave approximation can be assumed, and all the remaining oscillating terms are neglected, resulting for the complete Hamiltonian 2e , ĤI = � âe −iω t t + â † e iω t t σ+ e iω q t + σ− e −iω q t + be −iω n t + b † e iω n t σ+ e iω q t + σ− e −iω q t .

IQ mixer measurement scheme
In this Section we introduce the detection procedure, already explored in several other references 16,17 .By adding a drive with amplitude ε on the NEMS we can rewrite the effective Hamiltonian in interaction representation This Hamiltonian shows a beam-splitter-like interaction for phonon and photons, mediated by the qubit state population imbalance, plus the driving on the NEMS.This conditioned interaction allows the qubit to work as a switch for the NEMS and cavity excitations interchange.There are previous examples in the literature illustrating the use of a quantum switch 18 architecture to generate non-classical states of microwave radiation and to establish entanglement between the modes of the resonator and the degrees of freedom of the qubit.Here, the term "qubit" refers specifically to the qubit used for the operation of the quantum switch.
Let us consider the generation and transfer of superposition states and the entanglement between the resonators and the qubit.According to the Hamiltonian (9), given the initial state in the form where g is the ground state, and |e� is the excited state of qubit, the evolved state is written as where |α σ � and |β σ � are coherent states of the TLR and NEMS, respectively, for σ = g, e.
Therefore, with g| σz |g = −1 , e| σz |e = 1 , â σ = α σ , and b σ = β σ we have the following set of equations whose solutions are = sin(θ) g, 0 g , 0 g + cos(θ)|e, 0 e , 0 e �, (11)   www.nature.com/scientificreports/ The output of the TLR is directed to a microwave beam-splitter (BS), as can be seen in Fig. 1.The two outputs then pass through separate IQ mixers 16,17 .The four output currents from two IQ mixers can be correlated in various ways.Our particular used observable is the microwave field quadrature which gives direct access to all the normally ordered moments of the TLR field at the instant of measurement.The cross-correlations are calculated after the entire field has been detected and the TLR has returned to the vacuum state, ready for repetition of the same process.This results in the value of the quadrature signal measured at the TLR as With a simple differential analysis of the critical points, we can qualify the signal processing through the state of the quantum switch, where we have the maximum intensities at θ max = (8n + 3)π/4 , minimum at θ min = (8n + 7)π/4 , and no processing at θ 0 = (4n + 1)π/4 , for n = ..., −3, −2, −1, 0, 1, 2, 3, ... , all with the best detected quadrature Xφ=0 = X .The dynamical behavior is periodic, as shown in Fig. 3.

Fisher information associated with the quadrature measurement
The positive operator associated with a quadrature measurement is given by Notice that the results obtained in the measurement of a given quadrature are parameterized by a continuous variable x.Here is the value of φ that establishes the quadrature to be measured.The probability density of finding the value of a quadrature in φ with the value x is given by where the state |x φ � is defined as being the eigenstate of the quadrature Xφ , with eigenvalue x φ .The probability density P (ε|x) may be described as However, ψ α σ ≡ ψ α σ (x φ ) = x φ |α σ can be calculated by identifying that â|α σ � = α σ |α σ � , and so This is the same as the differential equation with the normalization Equation ( 19) has the solution resulting in a mixture of two Gaussian distributions of the form characterized by the driving force ε .The profile of P φ (ε|x) shows a distinct change of symmetry as θ is varied, as shown in Fig. 4, for φ = 0 and χt = π.The Fisher information associated with this measurement can be employed to infer the optimal parameters for inference of ε .It is calculated with the expression The numerical calculation of equation ( 22) in an integration in the interval between −30 ≤ x φ=0 ≤ 30 results in the graph in Fig. 5 with plots for different instants, showing good estimates of the force ε even for the situation with null average signal, as described in Fig. 3. Remark that depending on the behaviour of P φ (ε|x) , illustrated in Fig. 4, the superposition of the qubit state can directly influence the estimate of ε , at different times with a sudden or gradual increase in the LogPlot of F(ε, θ, φ) in relation to ε.

A quantum switch protocol
Given the optimal force estimation for θ = 3π/4 , we can extend the proposal to use the quantum switch to generate nonclassical states.For example, one could employ it to generate operations on the"qubit-like" states 25 encoded in superposition states of the TLR field 26,27 as and  Note that applying a new pulse to the NEMS can generate a CNOT logic gate, see Fig. 6.Other types of states can also be generated, such as GHZ 28 , and GKP states 29 .

Concluding remarks
In view of this, we conclude that the present hybrid circuit in the strongly dispersive regime presents a good means of processing the photon-phononic signal and an excellent detector through the quadrature measurement scheme using linear detectors.
The exploration of quantum switches and their role in entanglement phenomena are of remarkable relevance for quantum information processing 30,31 .The examples discussed in this context, particularly focusing on the force estimation, the creation of nonclassical states of microwave radiation, and the establishment of entanglement between NEMS, TLR and qubit, highlight the potential of quantum switches in manipulating quantum states for various applications.
The viability of implementing experimentally the present proposal is very high, considering that nearly a decade ago, in a similar circuit architecture 6 , it was possible to achieve values where = 120 MHz , = 1.35 MHz , δ n = −0.73GHz , δ T = 0.74 GHz , and that it has recently been shown that the devices involved can have dissipation rates of the order of a few kHz [32][33][34][35] .Previously, the implementation of the mixed IQ measurement for TLR was experimentally carried out in reference 36,37 .
The ability to exploit the quantum switch architecture for tasks such as coherent state transfer and the generation of entanglement provides a promising avenue for advancing quantum communication and computation technologies.These findings contribute to the growing body of knowledge in the field of quantum information science, offering insights into the controlled manipulation of quantum states at the quantum hybrid circuit level.

Figure 1 .
Figure 1.Schematic of a hybrid quantum circuit with IQ mixer measurements:The circuit schematized on a platform contains an electromagnetic shield with two grounded lines, and between them is a quantum hybrid circuit composed of the capacitive coupling between a TLR and a suspended NEMS mediated by a qubit superconducting2,5 .The TLR also has a channel capacitively coupled to its output that connects a beam splitter (BS) with two linear detectors for IQ mix measurements16,17 .